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feat: multiplication of intervals in rings #20636

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feat: multiplication of intervals in rings #20636

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37 changes: 36 additions & 1 deletion Mathlib/Algebra/Order/Interval/Basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -121,7 +121,7 @@ section Mul
variable [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α]

@[to_additive]
instance : Mul (NonemptyInterval α) :=
instance NonemptyInterval.instMul : Mul (NonemptyInterval α) :=
⟨fun s t => ⟨s.toProd * t.toProd, mul_le_mul' s.fst_le_snd t.fst_le_snd⟩⟩

@[to_additive]
Expand Down Expand Up @@ -529,6 +529,41 @@ instance divisionCommMonoid : DivisionCommMonoid (Interval α) :=

end Interval

namespace NonemptyInterval
variable [LinearOrderedSemiring α]

/-- Unlike `NonemptyInterval.instMul`, this works for semirings. -/
instance instMulOfSemiring : Mul (NonemptyInterval α) where
mul a b :=
{ fst := a.fst * b.fst ⊓ a.fst * b.snd ⊓ a.snd * b.fst ⊓ a.snd * b.snd
snd := a.fst * b.fst ⊔ a.fst * b.snd ⊔ a.snd * b.fst ⊔ a.snd * b.snd
fst_le_snd := by
dsimp
apply inf_le_sup.trans ?_
gcongr
apply inf_le_sup.trans ?_
gcongr
apply inf_le_sup.trans ?_
rfl }
Comment on lines +536 to +547
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This generalises the existing multiplication, right? Maybe we can merge both instances

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I think only under certain assumptions. Right now I don't think either satisfies the requirements of the other.

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Really? The existing one has [MulLeftMono α] [MulRightMono α], which implies a.fst * b.fst ⊓ a.fst * b.snd ⊓ a.snd * b.fst ⊓ a.snd * b.snd = a.fst * b.fst ⊓ a.snd * b.snd and similarly for ⊔


/-- `fst_mul` for rings. -/
@[simp] theorem fst_mul' (a b : NonemptyInterval α) :
(a * b).fst = a.fst * b.fst ⊓ a.fst * b.snd ⊓ a.snd * b.fst ⊓ a.snd * b.snd := rfl

/-- `fst_mul` for rings. -/
@[simp] theorem snd_mul' (a b : NonemptyInterval α) :
(a * b).snd = a.fst * b.fst ⊔ a.fst * b.snd ⊔ a.snd * b.fst ⊔ a.snd * b.snd := rfl

instance : MulZeroOneClass (NonemptyInterval α) where
mul_zero a := by ext <;> simp
zero_mul a := by ext <;> simp
mul_one a := by ext <;> simp [a.fst_le_snd]
one_mul a := by ext <;> simp [a.fst_le_snd]

-- TODO: Semiring (NonemptyInterval α)

end NonemptyInterval

section Length

variable [OrderedAddCommGroup α]
Expand Down
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